Sets of theorems with short proofs

1974 ◽  
Vol 39 (2) ◽  
pp. 235-242 ◽  
Author(s):  
Daniel Richardson
Keyword(s):  
Proof Complexity ◽  
Predicate Calculus ◽  
Definable Set ◽  
Semantic Tableaux ◽  
Skolem Functions ◽  
Definition Of

R. Parikh has shown that in the predicate calculus without function symbols it is possible to decide whether or not a given formula A is provable in a Hilbert-style proof of k lines. He has also shown that for any formula A(x1, …, xn) of arithmetic in which addition and multiplication are represented by three-place predicates, {(a1, …, an): A(a1, …, an) is provable from the axioms of arithmetic in k lines} is definable in (N,′, +,0). See [2].In §1 of this paper it is shown that if S is a definable set of n-tuples in (N,′, +, 0), then there is a formula A(x1, …, xn) and a number k so that (a1 …, an) ∈ S if and only if A(a1 …, an) can be proved in a proof of complexity k from the axioms of arithmetic. The result does not depend on which formalization of arithmetic is used or which (reasonable) measure of proof complexity. In particular, this gives a converse to Parikh's result.In §II, a measure of proof complexity is given which is based on counting the quantifier steps in semantic tableaux. The idea behind this measure is that A is k provable if in the attempt to construct a counterexample one meets insurmountable difficulties in the definition of the appropriate Skolem functions over sets of cardinality ≤ k. A method is given for deciding whether or not a sentence A in the full predicate calculus is k provable. The question for the full predicate calculus and Hilbert-style systems of proof remains open.

On the strong semantical completeness of the intuitionistic predicate calculus

1968 ◽  
Vol 33 (1) ◽  
pp. 1-7 ◽  
Author(s):  
Richmond H. Thomason
Keyword(s):  
Model Theory ◽  
Completeness Theorem ◽  
Predicate Calculus ◽  
First Order ◽  
Semantic Tableaux ◽  
Weak Completeness ◽  
Skolem Theorem ◽  
Completeness Theorems

In Kripke [8] the first-order intuitionjstic predicate calculus (without identity) is proved semantically complete with respect to a certain model theory, in the sense that every formula of this calculus is shown to be provable if and only if it is valid. Metatheorems of this sort are frequently called weak completeness theorems—the object of the present paper is to extend Kripke's result to obtain a strong completeness theorem for the intuitionistic predicate calculus of first order; i.e., we will show that a formula A of this calculus can be deduced from a set Γ of formulas if and only if Γ implies A. In notes 3 and 5, below, we will indicate how to account for identity, as well. Our proof of the completeness theorem employs techniques adapted from Henkin [6], and makes no use of semantic tableaux; this proof will also yield a Löwenheim-Skolem theorem for the modeling.

scholarly journals THE DEFINABILITY STRENGTH OF COMBINATORIAL PRINCIPLES

2016 ◽  
Vol 81 (4) ◽  
pp. 1531-1554 ◽  
Author(s):  
WEI WANG
Keyword(s):  
Combinatorial Problem ◽  
Reverse Mathematics ◽  
Ramsey's Theorem ◽  
Combinatorial Principle ◽  
Definable Set ◽  
Ramsey’S Theorem ◽  
Combinatorial Principles ◽  
Definition Of ◽  
Image Position

AbstractWe introduce the definability strength of combinatorial principles. In terms of definability strength, a combinatorial principle is strong if solving a corresponding combinatorial problem could help in simplifying the definition of a definable set. We prove that some consequences of Ramsey’s Theorem for colorings of pairs could help in simplifying the definitions of some ${\rm{\Delta }}_2^0$ sets, while some others could not. We also investigate some consequences of Ramsey’s Theorem for colorings of longer tuples. These results of definability strength have some interesting consequences in reverse mathematics, including strengthening of known theorems in a more uniform way and also new theorems.

An intuitionistically plausible interpretation of intuitionistic logic

1977 ◽  
Vol 42 (4) ◽  
pp. 564-578 ◽  
Author(s):  
H. C. M. de Swart
Keyword(s):  
Intuitionistic Logic ◽  
Predicate Calculus ◽  
Mathematical Treatment ◽  
First Order ◽  
The Past ◽  
Definition Of ◽  
Image Position ◽  
Order Predicate Calculus ◽  
Made In

Let IPC be the intuitionistic first-order predicate calculus. From the definition of derivability in IPC the following is clear:(1) If A is derivable in IPC, denoted by “⊦IPCA”, then A is intuitively true, that means, true according to the intuitionistic interpretation of the logical symbols. To be able to settle the converse question: “if A is intuitively true, then ⊦IPCA”, one should make the notion of intuitionistic truth more easily amenable to mathematical treatment. So we have to look then for a definition of “A is valid”, denoted by “⊨A”, such that the following holds:(2) If A is intuitively true, then ⊨ A.Then one might hope to be able to prove(3) If ⊨ A, then ⊦IPCA.If one would succeed in finding a notion of “⊨ A”, such that all the conditions (1), (2) and (3) are satisfied, then the chain would be closed, i.e. all the arrows in the scheme below would hold.Several suggestions for ⊨ A have been made in the past: Topological and algebraic interpretations, see Rasiowa and Sikorski [1]; the intuitionistic models of Beth, see [2] and [3]; the interpretation of Grzegorczyk, see [4] and [5]; the models of Kripke, see [6] and [7]. In Thirty years of foundational studies, A. Mostowski [8] gives a review of the interpretations, proposed for intuitionistic logic, on pp. 90–98.

scholarly journals On Languages Which are Based on Non-Standard Arithmetic

1963 ◽  
Vol 22 ◽  
pp. 83-117 ◽  
Author(s):  
Abraham Robinson
Keyword(s):  
Normal Form ◽  
Standard Model ◽  
Natural Number ◽  
Predicate Calculus ◽  
Natural Numbers ◽  
Standard Case ◽  
Logical Syntax ◽  
Skolem Functions ◽  
Concept Of Truth ◽  
Logical Calculi

The natural numbers play a part in the formulation of logical syntax inasmuch as they are used to count the symbols in a sentence, or the sentences in a proof, etc. In the present paper, we shall study an infinitary logical calculus which is based on replacing the ordinary natural numbers in the capacity just mentioned, by a non-standard model of arithmetic. (Compare refs. 3, 5, 6 for some other logical calculi of infinitary nature). Thus, our calculus will include formulae of length n for any natural number n, finite or infinite, in the chosen non-standard model of arithmetic. Evidently, the study of such formulae can be of interest only if we introduce concepts which are beyond the power of expression of the notions borrowed from the standard case. It turns out that the concept of truth in a model, when defined by means of Skolem functions has this character and involves a curious phenomenon which is analogous to one first pointed out by H. Steinhaus and J. Mycielski for another kind of infinitary language. Thus, while in the standard predicate calculus the negation of a sentence in prenex normal form is reduced to prenex normal form by changing the type of the quantifiers and by shifting the sign of negation, this procedure is not legitimate when truth is defined in this way in our calculus.

On the definition of an infinitely-many-valued predicate calculus

1960 ◽  
Vol 25 (3) ◽  
pp. 212-216 ◽  
Author(s):  
Joseph D. Rutledge
Keyword(s):  
Predicate Calculus ◽  
Extended System ◽  
Index Set ◽  
Formal Framework ◽  
Individual Variables ◽  
Definition Of ◽  
The Individual

This paper presents a class of plausible definitions for validity of formulas in the infinitely-many-valued extension of the Łukasiewicz predicate calculus, and shows that all of them are equivalent. This extended system is discussed in some form in [3] and [4]; the questions discussed here are raised rather briefly in the latter.We first describe the formal framework for the validity definition. The symbols to be used are the following: the connectives + and ‐, which are the strong disjunction B of [2] and negation respectively; the predicate variables Pi for i ∈ I, where I may be taken as the integers; the existential quantifiers E(J), where J⊆I, and I may be thought of as the index set on the individual variables, which however do not appear explicitly in this formulation.

Literacy and Ideology

1991 ◽  
Vol 12 ◽  
pp. 71-85 ◽  
Author(s):  
Elsa Auerbach
Keyword(s):  
Social Control ◽  
Social Order ◽  
Power Relations ◽  
Institutional Practices ◽  
Definable Set ◽  
The Past ◽  
The Social ◽  
Values And Beliefs ◽  
Ideological Perspective ◽  
Definition Of

The past decade has seen increasing acceptance of the perspective that there can be no disinterested, objective, and value-free definition of literacy: The way literacy is viewed and taught is always and inevitably ideological. All theories of literacy and all literacy pedagogies are framed in systems of values and beliefs which imply particular views of the social order and use literacy to position people socially. Even those views which paint literacy as a neutral, objectively definable set of skills are in fact rooted in a particular ideological perspective, and it is precisely because they obscure this orientation that they are most insidious. In fact, as Fairclough (1989) argues, one of the primary mechanisms of social control is the “naturalization” of institutional practices which legitimize and perpetuate existing power relations.

A Three-Valued Logic for Software Specification and Validation. Tertium tamen datur

1991 ◽  
Vol 14 (4) ◽  
pp. 411-453
Author(s):  
Beata Konikowska ◽  
Andrzej Tarlecki ◽  
Andrzej Blikle
Keyword(s):  
Natural Deduction ◽  
Deductive System ◽  
Predicate Calculus ◽  
Critical Survey ◽  
Inference Rules ◽  
Consequence Relations ◽  
Software Specification ◽  
Semantic Tableaux ◽  
Deduction Rules ◽  
Development And Validation

Different calculi of partial or three-valued predicates have been used and studied by several authors in the context of software specification, development and validation. This paper offers a critical survey on the development of three-valued logics based on such calculi. In the first part of the paper we review two three-valued predicate calculi, based on, respectively, McCarthy’s and Kleene’s propositional connectives and quantifiers, and point out that in a three-valued logic one should distinguish between two notions of validity: strong validity (always true) and weak validity (never false). We define in model-theoretic terms a number of consequence relations for three-valued logics. Each of them is determined by the choice of the underlying predicate calculus and of the weak or strong validity of axioms and of theorems. We discuss mutual relationships between consequence relations defined in such a way and study some of their basic properties. The second part of the paper is devoted to the development of a formal deductive system of inference rules for a three-valued logic. We use the method of semantic tableaux (slightly modified to deal with three-valued formulas) to develop a Gentzen-style system of inference rules for deriving valid sequents, from which we then derive a sound and complete system of natural deduction rules. We have chosen to study the consequence relation determined by the predicate calculus with McCarthy’s propositional connectives and Kleene’s quantifiers and by the strong interpretation of both axioms and theorems. Although we find this choice appropriate for applications in the area of software specification, verification and development, we regard this logic merely as an example and use it to present some general techniques of developing a sequent calculus and a natural deduction system for a three-valued logic. We also discuss the extension of this logic by a non-monotone is-true predicate.

operators and alternating sentences in arithmetic

1980 ◽  
Vol 45 (1) ◽  
pp. 144-154 ◽  
Author(s):  
Larry Manevitz ◽  
Jonathan Stavi
Keyword(s):  
Standard Model ◽  
Natural Number ◽  
Combinatorial Problem ◽  
The Other ◽  
Predicate Calculus ◽  
Completeness Proof ◽  
The Standard Model ◽  
Truth Value ◽  
The One ◽  
Definition Of

Determining the truth value of self-referential sentences is an interesting and often tricky problem. The Gödel sentence, asserting its own unprovability in P (Peano arithmetic), is clearly true in N(the standard model of P), and Löb showed that a sentence asserting its own provability in P is also true in N (see Smorynski [Sm, 4.1.1]). The problem is more difficult, and still unsolved, for sentences of the kind constructed by Kreisel [K1], which assert their own falsity in some model N* of P whose complete diagram is arithmetically defined. Such a sentence χ has the property that N ⊨ iff N* ⊭ χ (note that ¬χ has the same property).We show in §1 that the truth value in N of such a sentence χ, after a certain normalization that breaks the symmetry between it and its negation, is determined by the parity of a natural number, called the rank of N, for the particular construction of N* used. The rank is the number of times the construction can be iterated starting from N and is finite for all the usual constructions. We also show that modifications of, e.g., Henkin's construction (in his completeness proof of predicate calculus) allow arbitrary finite values for the rank of N. Thus, on the one hand the truth value of χ in N, for a given “nice” construction of N*, is independent of the particular (normalized) choice of χ, and we shall see that χ is unique up to (provable) equivalence in P. On the other hand, the truth value in question is sensitive to minor changes in the definition of N* and its determination seems to be largely a combinatorial problem.

Inductive definitions and proofs

2018 ◽  
Author(s):  
Vann McGee
Keyword(s):  
Negative Information ◽  
Predicate Calculus ◽  
Inductive Definition ◽  
Positive Information ◽  
Inductive Definitions ◽  
Closure Conditions ◽  
Definition Of ◽  
Special Case

An inductive definition of a predicate R characterizes the Rs as the smallest class which satisfies a basis clause of the form (β(x)→Rx), telling us that certain things satisfy R, together with one or more closure clauses of the form (Φ(x,R)→Rx), which tell us that, if certain other things satisfy R, x satisfies R as well. ’Smallest’ here means that the class of Rs is included in every other class which satisfies the basis and closure clauses. Inductive definitions are useful because of inductive proofs. To show that every R has property P, show that the class of Rs that have P satisfies the basis and closure clauses. The closure clauses tell us that if certain things satisfy R, x satisfies R as well. Thus satisfaction of the condition Φ(x,R) should be ensured by positive information to the effect that certain things satisfy R, and not also require negative information that certain things fail to satisfy R. In other words, the condition Φ(x,R) should be monotone, so that, if R ⊆S and Φ(x,R), then Φ(x,S); otherwise, we would have no assurance of the existence of a smallest class satisfying the basis and closure conditions. While inductive definitions can take many forms, they have been studied most usefully in the special case in which the basis and closure clauses are formulated within the predicate calculus. Initiated by Yiannis Moschovakis, the study of such definitions has yielded an especially rich and elegant theory.

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